Rigid Bodies

Abstract

The motion of a rigid body at any instant is determined by the six components of two vectors, the angular velocity ω and the velocity u of a chosen point O of the body. There are therefore six degrees of freedom and we should be able to describe the evolution by introducing six generalized coordinates: three for position and three for orientation.

The position coordinates are straightforward because we can use the three Cartesian coordinates of O in some inertial frame. The components of u are then the corresponding generalized velocities. As we saw in Section 1.9, it is a more complicated problem to find convenient coordinates to describe the rotational degrees of freedom. It is, in fact, impossible to find three generalized coordinates for which the three components of ω are the corresponding generalized velocities. Whatever angular coordinates are used, the task of expressing ω in terms of their time derivatives is always a source of complication.